NASA Contractor Report
ICASE Report No. 93-69
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SIMULATIONS OF DIFFUSION-REACTION EQUATIONS WITH
IMPLICATIONS TO TURBULENT COMBUSTION MODELING
Sharath S. Girimaji
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NASA Contract No. NAS 1-19480
September 1993
Institute for Computer Applications in Science and Engineering
NASA Langley Research Center
Hampton, Virginia 23681-0001
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National Aeronautics and
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Langley Research CenterHampton, Virginia 23681-0001
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ICASE Fluid Mechanics
Due to increasing research being conducted at ICASE in the field of fluid mechanics,
future ICASE reports in this area of research will be printed with a green cover. Applied
and numerical mathematics reports will have the familiar blue cover, while computer science
reports will have yellow covers. In all other aspects the reports will remain the same; in
particular, they will continue to be submitted to the appropriate journals or conferences for
formal publication.
SIMULATIONS OF DIFFUSION-REACTION
EQUATIONS WITH IMPLICATIONS TO TURBULENT
COMBUSTION MODELING
Sharath S. Girimaji 1
Institute for Computer Applications in Science and Engineering
NASA Langley Research Center
Hampton, VA 23681
ABSTRACT
An enhanced diffusion-reaction reaction system (DRS) is proposed as a statistical model
for the evolution of multiple scalars undergoing mixing and reaction in an isotropic turbulence
field. The DRS model is close enough to the scalar equations in a reacting flow that other
statistical models of turbulent mixing that decouple the velocity field from scalar mixing and
reaction (e.g. mapping closure model, assumed-pdf models) cannot distinguish the model
equations from the original equations. Numerical simulations of DRS are performed for
three scalars evolving from non-premixed initial conditions. A simple one-step reversible
reaction is considered. The data from the simulations are used (i) to study the effect of
chemical conversion on the evolution of scalar statistics, and (ii) to evaluate other models
(mapping-closure model, assumed multivariate fl-pdf model).
1This research was supported by the National Aeronautics and Space Administration under NASA Con-tract No. NAS1-19480 while the author was in residence at. the Institute for Computer Applications in
Science and Engineering (|CASE), NASA Langley Research (:enter, Hampton, VA 23681.
PI_E_D, tNG F'._.GE BLANK NOT FILMED ,..
111
1 Introduction
Turbulent combustion is an interplay of several complex physical processes: e.g. turbulent
transport, scalar mixing, chemical reaction, heat release. Turbulent transport is present
only in inhomogeneous flows where the gradients of the scalar statistics are non-zero. Scalar
mixing- which is important in isotropic as well as inhomogeneous flows- comprises two mech-
anisms: the effect of the velocity field, which is to cascade scalar energy to smaller length
scales; and molecular diffusion, which typically acts at smaller scales to reduce scalar fluctua-
tions. Chemical reaction is a local phenomenon which converts scalar species from reactants
to products. In problems of interest in combustion, chemical conversion is accompanied by
heat release and, hence, density reduction. If the heat release is large enough, the resulting
buoyancy forces can modify the velocity field in which the scalars reside If the heat release
is small, the modification of the velocity field is negligible.
In the scalar-statistics equations, the terms representing several of the above processes
need closure modeling. A turbulent combustion model is really a composite of these closure
models of the individual processes. Often, when calculations from the composite model are
in poor agreement with either experimental or direct numerical simulation (DNS) data, it
is difficult to isolate the underlying cause (the model of a specific physical process) of the
disagreement. For this reason, it is useful to construct sample test problems where only one
or few of the physical processes are important (other processes are either negligible or or
completely absent). Data from the numerical simulations of such a test problem call be used
to construct and validate the individual models.
The divide-and-conquer approach has proved especially successful in the understanding
and modeling of the inert scalar mixing process. Eswaran and Pope (1988), and subsequently
others, performed DNS of inert scalar fields evolving from non-premixed initial conditions in
constant-density, isotropic turbulence. In this first step, the effect of velocity field on scalar
mixing is studied, without the complexities of spatial inhomogeneities or chemical reaction.
One of the chief findings of DNS studies is that (for the range of Reynolds number inves-
tigated, R_ 30 - 90) the velocity field only affects the timescale of the scalar pdf evolution.
In appropriately normalized time, the scalar pdf evolution is independent of the velocity
field. This insight gained from the DNS studies spawned reasonably successful modeling
approaches in which the scalar evolution is considered only in the normalized time, thus
decoupling tile velocity field from the scalar field. The models are: the mapping closure
model (Kraichnan 1989, Girimaji 1992b), the Johnson-Edgeworth translation (JET) models
of Madnia et al (1992) and the assumed/3-pdf model (Girimaji 1991a).
Study of a test case which combines scalar mixing with chemical chemical reaction in
isotropic turbulence should be a useful second step towards the ultimate objective of un-
derstanding and modeling turbulent combustion. In this work, we analyze the test-case
equations and propose a statistical model for the scalar (species and temperature) evolu-
tion. The model equation set is a diffusion-reaction system (DRS). The model accounts for
tile effect of the velocity field via a time-dependent diffusion enhancement coefficient factor.
The model equations are close enough to the original turbulent-reaction equations that other
models which decouple the velocity field and scalar-field mixing cannot distinguish between
the two sets of equations. Numerical simulation of the model DRS equations is computation-
ally much less intensive than DNS of the original equations and may.serve nearly as useful a
purpose in understanding and evaluating turbulent combustion models. Simulations of the
DRS equations are performed to (i) shed some light on the effect of reaction on scalar field
evolution, and (ii_ evaluate the performance of other modeis(devel0ped originally for inert
mixing) against reacting scalar data.:
The organization of the remainder of the paper is as follows. In Section 2, the relevance
of the (enhanced) DRS to turbulent combustion is investigated in more detail. Numerical
simulations of DRS equations is performed in in section 3. A brief description of the numerics
employed in the simulations is also provided. Section 4 contains the results of the simulation
and a discussion of its implications to turbulent combustion modeling. The paper concludes
in Section 5 with a summary.
27
2 Enhanced diffusion-reaction system as a model of
turbulent reactive flows
Consider the mixing and reaction of n scalars (of mass fractions ¢_, a = 1,-..n) in a
velocity field u(X, t) evolving under Navier-Stokes. The following assumptions are made: i)
no body force, ii) no radiative heat transfer, iii) scalar diffusion governed by Fickian law, iv)
low Mach number (hence, pressure is considered nearly a constant), and v) high Reynolds
number (hence, the viscous contribution to the energy equation is negligible). Subject to
the above assumptions, the scalar and temperature (T) evolution equations are
,a¢, a¢_ a " D 0¢_, (1)
and
= - h_ _. (2),or.-5-i- + u,b-Zl _,[ 8-_,] + _ _a=l c_=l
In the above equations, D_ and w_ are respectively the Fickian diffusion coefficient and
the chemical production rate of species c_. The heat conductivity of tile mixture is A. The
enthaly of formation and thermal enthaly of species ol are h ° and h T. The total thermal
enthalpy of the mixture, (hT), is given by
= ,ohm. (3)c_=l
The pressure (p), density of the mixture (p) and the mass fractions are related through the
equation of state which could be the ideal gas law. It is assumed that the temperature and
the chemical source terms can be determined given the thermal enthalpy and mass fractions.
Subject to homogeniety of the turbulent velocity field, the joint pdf of scalar composition
and thermal enthalpy, F(h T, _¢), evolves according to (Pope 1985)
p(hT,¢)OF(_,¢) 0_ - OhT[F(T,¢_){07,- RT}] (4)
o_=1
In the above equation the conditional diffusion of temperature (OT) and scalar (@¢_) are
given by
a ,_aT r _]IT,_¢), (5)o_ = (_-kT[_ + _ ph.D_o,,'ct=]
3
an d
• 0 "D O¢:'IT,_, (6)°+o = (b-_,t :_,,J
where, the notation (b]r) denotes the conditional mean of b with respect to r. The pdf source
term of temperature (RT) and scalar (Ro_) due to reaction are given by
T'g
RT _ ___ o= hawa ,
and
a=l
(7)
n,o = w.. (8)
While the source terms due to reaction are closed in terms of temperature and scalar compo-
sition, the scalar diffusion terms need closure modeling. The modeling of these terms require
that the effect of the velocity field on the scalar field be known, at least approximately.
2.1 Lagrangian coordinate analysis
The effect of the velocity field on tile scalar field is examined in Lagrangian frame of reference
(x, t) which is cartesian. For the sake of this analysis, the velocity field is assumed to be
isotropic. The Eulerian coordinate X is now nonstationary, curvilinear and nonorthogonal,
evolving according to
OX(x, t)ot - u[x(x, t), t]. (9)
The Eulerian conditional diffusion can now be written in terms of the Lagrangian deriva-
tires:
Define
O 0¢_, Ox_ Oxb a [D 0¢.] (10)
OXa
C_,(x) = _._. (11)
The correlation between Cia and the scalar derivatives in the Lagrangian coordinates is
discussed in detail in Girimaji (1992 c) for the case of inert scalar mixing. It is pointed out
that Ci_ related to the rate at which the turbulence stretches a material surface attached
to x. The determinant of Ci_, tel, is likely to be larger than unity if the material surface
is stretched by turbulence and smaller than unity if the surface is shrunk. On an average
4
turbulence stretches material surfaces leading to ]C] being larger than unity. This results in
the Eulerian scalar derivatives (following a fluid particle) to be larger than ones in Lagrangian
coordinates. The length scale of variation of Ci_ is likely to be of the order of the characteristic
length scale of the small scales which cause the strecthing or shrinking. This scale is the
Kolmogorov length scale, 77. In the absence of reaction, the length scale of variation of the
Lagrangian scalar derivatives will depend largely on the initial length scale of the scalar
field. It was argued in M Girimaji (1992c) that, if the Kohnogorov length scale is of a
different order of magnitude than the initial length scale of the scalar field then, Ci_ and the
Lagrangian scalar derivatives would be poorly correlated.
Chemical reaction can cause the Eulerian scalar derivatives to be different from those in
inert mixing. It is important to evaluate the validity of the poor-correlation simplification
in the reacting case. In the distributed regime of turbulent combustion (DamkShler number
smaller than unity), the reaction zone (flame thickness) is larger than the Kolmogorov length
scale. It is unlikely that chemical reaction can cause the scalar derivatives to be significantly
different from the inert-mixing case. As a result, as in the case of inert mixing, the Lagrangian
scalar derivatives would be poorly correlated with Cir. In the flamelet regime of turbulent
combustion (DamkShler number greater than unity), the flame thickness is smaller than the
Kolmogorov length scale. Infact, at the limit of infinite DamkShler number, the reaction zone
can be infinitesimally small, causing a discontinuity in the scalar fields. There is no chemical
reaction outside these discontinuities. In the reaction zone, it is not clear if the Lagrangian
derivative will be poorly correlated with Ci=, whereas, outside that zone (regions of no
reaction) the poor-correlation simplification is as valid as in the inert-mixing case. Inside
the reaction zone the chemical source term is dominant and the molecular diffusion term-
and hence its modeling- is unimportant. With increasing DamkShler number - decreasing
flame thickness - the volume of the flow field over which the simplification is valid increases
and the importance of the diffusion term inside the reaction zone diminishes further. In the
limit of DamkShler number going to infinity, the simplification is valid almost everywhere
in the flow field except at the inifinitesimally thin flame sheet where the molecular term is
insignificantly small. As a result, the poor-correlation simplification should yield reasonable
results even in the flamelet regime of combustion, especially at large Damk6hler number.
Hence, the expression for the conditional diffusion can be simplified as
.OXa OXb 0 00_ b OX_ OXb](___ a O¢a]]hT 'O,_= (-ff-_OX, Ox [PD_ ]]hT, C_) _ (OX, OX ,, [pD_-_x b ¢_). (12)
The quantity / °-_ °-_-\ is a second-order tensor which, if the turbulent field is isotropic, can\ OXi OXi /
be written as
1 o_ o__where S(t) = 5 ax. ox_"
Ox,, Ozb _"OY,,aX," = (S(t))_ob
Hence the model for O¢,, is
Oct(T, ¢)= (S(t))(O-_[D,_O¢_]IhT,¢).-- _ -
Using similar arguments the model for O T can be derived:
( O___[aOT T 0¢.1OT= (S(t)) Ox_ Oza+ _=,_ ph.D.-_z jIhT,_,
(13)
The model.
diffusion be modeled by
(14)
(15)
and
Based on the Lagrangian frame analysis, it is proposed that conditional
fi Tm O¢'_ ]lh T,_+ ph. D_-o-_x j , ,¢_), (16)
_=1 a
(_L_0[_OT"OT= (S(t)) 0_o Ox_
(17)
dhTm ,,_- (S(t))O--_--[)_ Or + fi h Tmn 0Ca' fi h°w_(¢m). (19)
P -_ Ox_ Oz. _=1p_ v_-b-_x_J- =1
d¢2 o 0¢2P--d'[- = (S(I))-_x_[PD_-_x_ ] + w_(C-m)' (18)
an d
In the above equation, the superscript m refers to the model variables.
The effect of the velocity field on the scalar evolution in the Lagrangian reference-frame
is much smaller than the effect in the Eulerian reference frame. Under the poor-correlation
simplification, the scalar field evolution in the Lagrangian frame can be considered indepen-
dent of the velocity field. Hence, the model variables (h Tm and ¢m) can be considered to
evolve according to
Equations (18) and (19) have two important uses. First, they form a statistical model of
the chemically-reacting flow equations (1) and (2). Second, the model equations are similar
enough to the original equations that many of the other scalar mixing models - multiscalar
mapping closure models (Pope 1991, Gao and O'Brien 1991, Girimaji (1993)), multivariate
fl-pdf model, JET model - cannot distinguish the difference between pdf's of the two sets of
instantaneous equations. Hence, for validating many statistical aspects of the other models,
DNS of equations (18) and (19) can be used rather than the more expensive DNS of turbulent
reacting flow equations (1)and (2).
What is (S(t))? The quantity (S(t)) is a measure of the mean deformation of the Eule-
rian coordinate frame caused by the velocity field and can be interpreted as the diffusion
enhancement factor. It can be expressed in terms of the deformation characteristics of a
material cube attached to a Lagrangian point x (Girimaji 1992c):
1 2S(x,t) = _--_[Al(t ) + Ai(t ) + A](t)], (20)
where A.(x, t) is the area of surface of the cube which was initially coincident with tile a
axis and V(x, t) is the volume of the cube. The initial values are such that,
A,(x, 0) = A2(x,0)= A3(x, 0)= V(x,0)= S(x, 0)= 1.
For an incompressible velocity field, the volume V(x, t), which also represents the density
ratio following a fluid particle, is always unity. The compressibility of the velocity field
manifests itself on the scalar mixing through V. In an isotropic velocity field, the various
area magnitudes (A1, A2, and A3) are all statistically equivalent. Hence, assuming isotropy
(and incompressibility)
(S(t)) = (A2(x,t)), (21)
where A(t) represents the area at any time t of any typical material-surface element (of initial
unit area) associated with the fluid element. Material element deformation in isotropic tur-
bulence was studied by Girimaji and Pope (1990) and its implications on (S(t)) is discussed
7
in detail by Girimaji (1992c). It suffices to say here that in isotropic turbulence (S(t)) grows
ill time nearly exponentially. The logarithmic growth rate is approximately
dln<S) 0.8p( t ) -- ,_ --, (22)
dt %
where To is the Kolmogorov time scale of the turbulence. The diffusion enhancement factor
(S(t)} reflects the cascading effect of the velocity field on the scalar field. As the scalar
energy cascades from large scales to small scales, the Eulerian scalar gradient following a
fluid element increases exponentially. The diffusion enhancement effect of the velocity field
is more prominent at later times when more of the scalar energy is in the small scales.
3 Further simplification and simulation
We want to consider the simplest non-trivial case of multiscalar mixing and combustion
where reaction affects the evolution of scalar pdf. For this purpose, several previous authors
(e.g., Pope 1991) have used the following set of equations:
+ u,==.on,- "_ o_,ox, + wo. (23)
The above equation represents constant-density, isothermal chemical reactions between scalars
of different, but, constant diffusivities. If the velocity field U is isotropic, the enhanced-
diffusion reaction model corresponding to equation (23) is
d¢_ 2.,_ (S(t))D_o_oz" + w_(¢_m). (24)
The time-dependent diffusion enhancement factor is calculated from equation (22).
We consider reversible reactions of tile type
A+B_2P, (25)
where, scalars A and B are reactants and scalar P is the product. The factor two is included
in equation (25) to conserve mole fractions. The chemical production rate of the three species
are
WA = --]_]¢ACB "4- _¢b(_2c (26)
ws = --kj¢ACB + kb¢_
wp = 2k]dACs --' 22kb¢c,
8
where, k! and kb are the forward and the backward reaction-rate coefficients. The backward
coefficient" is usually expressed in terms of the forward rate: kb = kek 1, where ke is the
equilibirium constant.
3.1 Simulations
The problem of non-premixed, randomly and isotropically distributed scalars undergoing
enhanced mixing and chemical reaction is considered. Numerical simulations of equation
(24) are performed in a cubical domain with periodic boundary conditions. A pseudo-
spectral method is used: the scalar spatial derivatives are sought in spectral space, whereas,
the reaction-rate term is calculat'ed in physical space. The initial scalar field is specified in
the manner described in Girimaji (1993). The initial scalar field has a prescribed length
scale l¢, and the initial joint pdf, F(¢A, ds), is given by
F(¢A, CB;0)= #A(0)5(¢A -- 1)5(¢B)+ ftB(O)t_(+A)t_(dPB- 1) + _p(O)5(¢A)5((OS). (27)
In the above equation, #A, /_B and #p represent the means of species A, B and P respectively.
The mass fraction of P is given by
Cp = 1 - CA -- CB" (28)
Even when the scalar field evolution is governed by the simple enhanced-diffusion reaction
system (equations 24 and 26), the scalar joint-pdf evolution is affected by several parameters:
i)ro (the Reynolds number); ii) l¢ (initial length scale of the scalar field); iii) DA, DB
(Prandtl number and relative diffusion); iv) _a, #e (stoichiometric ratio); v) ks (Damk5hler
number); and, vi) kb (reversibility). In this study, we fix the values of r, 7 (corresponding
to a Taylor-scale Reynolds number of approximately 60, as reported in Girimaji and Pope
1990). The simulations are divided into four groups based on the initial joint pdf. Within
each group several simulations are performed by varying the other parameters. The values
of the parameters for the various simulations are shown in Table 1.
4 Results and implications to turbulent combustion
modeling
Tile results of the simulations are presented in two parts. First, the effect of reaction on
the evolution of the means, variances, pdf's and other quantities of interest are studied.
Secondly, the simulation data are used in its role as the facsimilie of DNS to evaluate two
models of turbulent combustion: the mapping closure model and the assumed/3-pdf model.
The effect of chemical reaction on the following statistics of various species are examined:
1. The mean, variances and correlations.
2. The evolution of the pdf.
3. The mean scalar dissipation and Chemical production rate of variance.
4. The evolution of the conditional scalar dissipation.
Results from the simulations of Group 1 and Group 2 are used in this part of the study.
Statistics of species A and P only are presented, since ¢B can be completely determined
knowing ¢A and Cp. In the discussions below, angular brackets imply mean and primes
denote fluctating part of a random variable.
:i
Means. The mean evolutions for Groups 1 and 2 are presented in Figures 1 and 2 respec-
tively. As is to be expected, the means do not change for the inert cases: (Ca) remains at its
initial value, and (¢p) is always zero. As for the reacting cases, the mean reactant decreases
more rapidly for larger kI (reaction rate coefficient) than for smaller k I. In the reversible
reaction cases (Clb and C2b), the mean values asymptote to equlibrium values given by the
equation
A.B = (29)
The above equation in conjunction with equation (28) leads to the following equlibrium
values: for case Clb,/tA = 0.33, /tB = 0.33, #p = 0.33; and for case C2b, PA = 0.116,/_B =
0.616, _p = 0.267.
=
10
Variances. The evolution of variancein reacting field in given by
da---_2= 2(-_ + _:,), (30)dt
where, the mean scalar dissipation (e_) and the production/destruction of variance due to
d ,0¢" 0¢'_, (31)
/ !r = (w.¢_). (32)
reaction (e_) are defined as
The variances of species A for Groups 1 and 2 are shown in Figures 3 and 4. In Group 1,
reaction appears to cause the variance to decay slowly compared to the inert case. On the
contrary, in Group 2, the variance decays more rapidly with reaction than without. The effect
of reaction on Group 2 is larger than that on Group 1. The reasons for these observations
are explained further below, when the behavior of _4 is discussed. Although discernable,
the magnitude of the modification due to reaction is small enough to be unimportant for
practical problems. As shown in Figures 6 and 7, eI >> c_4 for the cases considered resulting
in only a slight modification of the variance evolution.
The variance of the product is initially zero. It grows in time initially, due to the chemical
production term; more rapidly in Group 2 than in Group 1. After attaining a peak value, it
diminishes due to effect of molecular action.
Correlation between CA and CB. The correlation coefficient between species A and B,
CAB = (¢AdB) (33)
is given in Figure 5 for the inert cases C2 and C4 and their reacting counterparts C2b and
C4b. At early times, when chemical conversion is small, correlation is not affected much
by chemical reaction. With time the correlation for the reacting cases deviate from their
inert counterparts, going to lower magnitudes while still preserving a negative sign. The
deviation, however, is not too large.
11
Scalar pdf. In order to examine the effectof reaction on the scalar pdf evolution, the pdf
of scalar A is plotted in Figure 6 for cases C1 and Clb. Figure 7 contains pdf's of scalar A
for cases C2 and C2b. Initially, chemical reaction does not affect the pdf much, for, given the
segregated initial condition, molecular mixing should occur first before chemical conversion.
However, at later stages, as chemical conversion becomes prevalent, the pdfs are strongly
affected. The pdf's develop positive skewness due to chemical depletion, and in both cases
tend to g-functions at zero at long times.
The pdf of the product, scalar P, is plotted in Figure 8 (Cases Cla and Clb) and Figure 9
(Cases C2a and C2b). The pdf's of both groups display bimodal behavior initially. One mode
of high probability is at zero value of mass fraction, representing parts of the field containing
umnixed reactants. The second mode of high probability is close to the maximum value of
mass fraction at that time. With time, the mode at zero disappears, and the other mode
migrates to higher values retaining its spike-like form, in cases Cla and Clb. The asymptotic
state of the pdf for Group 1 is a g-function at unity, representing a complete conversion of
all reactants to products. In cases C2a and C2b also the mode at zero dissipates with time.
The pdf migrates to the right and has wider support (larger variance) unlike Cla and Clb.
The asymptotic form of the pdf in these cases (Group 2) is a g-function at Cp = 0.5.
Chemical productlon-rate of variance. The chemical production-rate of variance of
species A is given by
r <¢A¢A¢_>"'A <w_¢'A) <CA> ' ' ' ' ' ' '= (¢_)<¢ACA= - <¢ACS>- >- (34)
For Groups 1 and 2, the correlation between A and B is nearly negative unity leading to
¢_ _-¢_, (35)
which when substituted in equation (34) leads to
(36)
Similar arguments lead to
(:37)
12
In Figures 10and 1I, e_of Groups 1and 2 are presented.The behavior of e_, in the two
groups can be understood by recalling that for Group 1, (phiA) = (¢B) and for Group 2,
(¢B) = (CA} + 0.5, at all times. This yields
4 (38)
for Group 1. As mentioned earlier, the pdf is positively skewed leading to a positive value
for e_. For Group 2
era _ -0.5(phil) + (¢_). (39)
For this group, the first term on the right hand side (of equation 39) dominates the skewness
term leading to a larger negative value of e_. The behavior of the variances of the two groups
is consistent with the above explanations.
Figures 12 and 13 contain e_, of groups 1 and 2. Both groups exhibit similar behavior.
Starting from zero, the value of e_o increases to a peak value and then diminishes to near-zero
values. The peak value is higher for higher k]. The early behavior of e_ of the reversible
case is identical to its non-reversible counterpart. However, its peak value is not as high,
and it diminishes much more rapidly to zero.
Mean scalar dissipation. The mean scalar dissipation of species A of Groups 1 and 2
are given in Figures 10 and 11. Similarly to its effect on variance, reaction causes the mean
scalar dissipation to decay more slowly in Group 1 and more rapidly in Group 2. Again,
the magnitude of modification is not very large. In both groups, the magnitude of the mean
scalar dissipation is much larger than that of chemical production-rate of scalar variance.
The mean scalar dissipation of P of Groups i and 2 are given in Figures 12 and 13. In
both the groups, with the formation of products, the mean scalar dissipation increases from
a zero initial value. After attaining a maximum value, the dissipation decreases. The mean
scalar dissipation achieves higher values in Group 2 than the corresponding cases in Group
1. Initially, e) is larger in magnitude than @, leading to a growth of a_, from its initial zero
value. At latter times, @ is larger leading to a decay in the variance levels. The implication
is that initially the scalar-P evolution is reaction controlled, after which both reaction and
diffusion are equally important, and ultimately the evolution is diffusion controlled.
13
Conditional scalar dissipation. The evolution of the conditional scalar dissipation of
scalar A in casesC2 (inert) and C2b (reacting) is given in Figure 14. Chemical reaction
has two effectson this quantity. The first is due to the more rapid declinein the maximum
valueof q_A as a result of reaction. The zero of the conditional dissipation migrates with the
extremum value (Girimaji 1992b). When mixing is accompanied by reaction, the conditional
dissipation is non-zero over a smaller range of q_a, than in the case of inert mixing. This
effect is negligible at the early stages and more pronounced in the later stages. The second
difference is the higher value of the conditional dissipation (where it is non-zero) at a given
value of mass fraction in the reacting case as compared to inert case. Again, this difference
is large in the latter stages and negligible in the early part. Despite the higher values of
the conditional scalar dissipation in the reacting case than in the inert case, the mean scalar
scalar dissipation is indeed lower in the reacting case, as shown in Figure 11. The reason for
this is the shift in the pdf (Figure 11) due to reaction. In the reacting case, the high values
of conditional dissipation occur at values of mass fraction of low probability of occurence
and vice versa. Whereas, in the inert case high probability and high conditional dissipation
appear to occur at nearly same values of mass fraction.
The conditional scalar dissipation of the product, scalar P, is given in Figure 15 for cases
C2a and C2b. The conditional dissipation of the product is very unlike that of the reactant
for this case. Initially, it is non-zero over a very narrow range of _bv values and it gradually
widens with time (or reaction). The peak value normalized by the mean scalar dissipation
decreases with time, indicating gentler gradients in the product field at later times. The
behavior of this quantity is further discussed later when the modeling issues are examined.
So far in this section, the effect of reaction on various quantities of interest was examined.
It is found that although the mean and pdf of the scalars are strongly affected by reaction,
other quantities like the variance (and, perhaps, other even moments) are not very different
from their inert-mixing counterparts. Hence, it would be useful to directly compare and
evaluate models of inert mixing against the enhanced-diffusion/reaction data.
14
4.1 Models vs. Data
We attempt to answer three specific questions regarding turbulent combustion modeling
using other models:
1. Can the inert mixing multiscalar mapping-closure model (Girimaji 1993) be used for
reacting flows without modification?
2. How good is the multivariate assumed/3-pdf model (Girimaji 1991a) for calculating
reacting flows?
3. How does the computationally simple assumed-pdf mode] compare with mapping-
closure model for multiscalar mixing?
Multiscalar mapping closure model. The multiscalar mapping closure model for inert
mixing (Girimaji 1993) is based on the simplification that tile conditional scalar diffusion of
a given scalar is a function of that scalar only:
a a¢,]lT, ¢)_ . a "D_j,O¢"I'TO¢_ = (_-_[D_ _-_i .._ (_-_i [ ,¢,). (40)
The conditional scalar diffusion for each scalar is obtained by employing the mapping closure
procedure for single scalar mixing. Knowing the conditional scalar diffusion of each scalar,
the joint pdf evolution is solved, and the results are in reasonably good agreement with data.
Even when mixing is accompanied by reaction, it is suggested in Girimaji (1993) that
the simplification stated in equation (40) may be valid. In the reacting case, however, tile
mapping closure procedure even for single scalar is not clear. So it will be useful to know, if
the conditional scalar diffusion implied by the mapping closure procedm'e for inert mixing is
adequate for the reacting case also. Conditional scalar diffusion is related to the conditional
scalar dissipation according to
1 0x¢_F(¢.) (41)O¢_= F(¢_) 0¢4
The inert case mapping closure model (with Gaussian reference field) for conditional scalar
dissipation for initially non-premixed reactants is (Girimaji 1992b)
X;(¢) _ exp(_2[crf__{2¢_ 1}]'2). (42)_:(o.5)
15
This model is quite good in the early stages of inert mixing and is technically invalid but
still adequate during the latter stages. The validity of the above model for the reacting case
is now investigated.
Shown in Figure 16, are F(¢A)X(¢A)/X(C_A = 0.5) of the data and model for case C2b.
In the model calculation of the above quantity, pdf F is taken from the data. The model
agrees with the data very well at early times. At later times, the model is still adequate.
Recall that even for the case of inert mixing the model is not very good at the final stages.
Perhaps, an inert-case model which is uniformly valid at all times would be good for reacting
case at all times too. In any event, for combustion applications, the behavior of the model
at the early stages, when the unmixedness is high, is more important than at the late stages
when the scalars are more or less uniformly mixed. Hence, for the reactants a closure model
for the conditional scalar diffusion obtained by substituting equation (42) in equation (41)
might be adequate.
Modeling the conditional diffusion of the product (scalar P) is not as simple. The relative
importance of chemical conversion and mixing in the scalar pdf evolution (of P) at various
times can be surmised from Figure 9 where the mean scalar dissipation and the chemical
production rate of variance are compared. Initially, the evolution of the scalar P is dom-
inated by chemical conversion. Molecular diffusion does not play a significant role in the
pdf evolution until much later. Therefore, it is important that the model for the conditional
diffusion (or dissipation) of the product be accurate at later times; accuracy at early times is
not as crucial. Comparison of F(¢p)X(Op)/X(¢p ---- 0.5) for the product is shown Figure 17
• for case C2b. The agreement is very poor in the initial stages. At the later stages, it is much
better. Whether the model is good enough for engineering applications can be determined
only from the computation of the pdf evolution using the closure model. Such a computation
is outside the scope of this work.
Assumed multivariate fl-pdf model. In the assumed-pdf approach the scalar joint pdf
is prescribed knowing the first few moments of the scalar field. It is computationally far less
intensive than the mapping closure model and is ideally suited for engineering computations.
16
The assumed multivariate/3 pdf model for a N-scalar mixing process is given by (Girimaji
1991a)
F(t_) = F(fl -]- " " " 3v flN)AB'-lA_32-1'''flpflNN-16(1 --¢1- "" -- CN)" (43)r(/31)-:The parameters of the model (fl,
turbulent scalar energy Q:1-S'
= q
In the above equation Q and S are given by
N N
• ", fiN) are functions of the mean mass fractions ,u_ and
sQ = Z: o, =
1). (44)
(45)a----1 a----1
2 is the variance of mass fraction of scalar a. The above model tested for two-scalarwhere o"a
inert mixing process with reasonable success, but is yet to be validated for multiscalar mixing
and reaction.
In Figure 18, the individual variances and some cross moments of species A and B cal-
culated using the model are compared against corresponding simulation data for case Clc.
Similar comparisons are performed for cases C2c, C3b and C4b in Figures 19, 20 and 21,
respectively. The means and the turbulent scalar energy required as inputs to the model are
taken from DRS data. The model predicts the variances reasonably well. The magnitude of
the cross covariance is consistently underpredicted by the model, but the difference is not too
much. For the two higher order cross moments compared ((¢_4_) and (4_¢_)) the model
does surprisingly well. For each of the cases considered, the moments behave differently and
the model is able to capture the behavior well, qualitatively and quantitatively. Overall the
performance of the model is quite satisfactory, given the simplicity of the model.
Multivatiate f-pdf model vs. Mapping closure model. Now tile assumed f-pdf
model is compared against the more detailed and computationally intensive mapping closure
model. The two models are equally good for the case of inert non-premixed two scalar
mixing (Girimaji 1992b). Comparison of the two models for multiscalar mixing has not
been performed before, and is attempted presently for inert mixing. (As mentioned before,
computations of multiscalar reacting flows using mapping-closure model are intensive to be
17
attempted here.) For the purpose of computation, simulations C3 and C4 are used. Ill the
absence of reaction, Groups 1 and 2 reduce to two-scalar mixing, and hence not useful to
evaluate multiscalar. In Figure 22, various normalized cross moments (of species A and B)
obtained from the simulations are plotted as a function of the variance of scalar A for Case
C3. (The variance decreases monotonically in time and hence can be used in lieu of time
as the abcissa.) Also shown in the Figure are the cross moments calculated using the two
models. The moments of the normalized variables are presented:
0_ -- (46)
For all the moments considered, both the models agree equally well with the data. Figure
23 presents the same comparison for case C4. In this case, the performance of the mapping
closure model is clearly superior. These findings are in keeping with the arguments presented
in Oirimaji (1991a) that the assumed fl-pdf model is likely to be more accurate when the
length-scale and diffusivities of the scalars are similar (case C3) than when they are widely
disparate (case C4). For case C4, even the mapping closure model is not very accurate. This
leads to the question, for engineering calculations, is the increase in accuracy achieved using
the mapping closure model worth the extra effort involved in computing? The answer will
depend on the application for which the model is being used. In combustion calculations it
is the unnormalized moments that appear in the scalar moment equations and need closure
modeling, not the normalized moments. Despite the relatively poor agreement of the nor-
realized moments in Figure 23, the unnormalized moments calculated using the/_-pdf model
are quite close to the data as seen in Figure 21 for the corresponding reacting case C4b.
This appears to suggest that the assumed-pdf approach, although simplistic, may be quite
adequate for modeling of turbulent combustion in engineering calculations.
5 Conclusion
It is shown that in isotropic turbulence the (enhanced) diffusion-reaction system (18 and
19) can be considered a statistical model of turbulent combustion (equations 1 and 2) with
the initial field and the diffusion enhancement factor ((S(t))) as inputs. The diffusion-
enhancement factor can be found if the scalar variance evolution is known (Girimaji 1992@
18
In the event that scalar varianceevolution is not known from any other source, a simple
model for the diffusion-enhancement factor is provided (equation 22). The enhanced-diffusion
reaction model is close enough to the turbulent combustion equations that, in normalized
time, many of the other models (mapping closure models, assumed-pdf models and any
model that decouples the velocity and scalar fields) cannot distinguish between the present
model and the original equations.
Simulations of the diffusion-reaction system is performed in a cubical box (64 z grid points)
from non-premixed initial conditions. The reaction considered is of the type
A+B_P.
The reaction is constant-defisity and cold (isothermal). Several simulations of different initial
conditions, reaction-rate coefficient and degree of reversibility are performed (Table 1). The
simulations are used to examine the effect of reaction on the evolution of several scalar
staistics (Figures 1 - 15). The data are also used to evaluate other models; mapping closure
model and the assumed multivariate/3-pdf model. The observations and inferences from the
study are the following.
1. The conditional scalar dissipation implied by the inert mapping closure model is rea-
sonable even for reacting case but only for the reactants (scalars A and B) as shown in
(Figure 16). For the product (scalar P), the agreement is quite poor at early times but
better at later times (Figure 17). Since, the evolution of scalar P is initially dominated
by chemical conversion and not molecular mixing, the accuracy of the conditional dis-
sipation model at the early stages may not be critical to the overall performance of the
model. At the later stages, when the evolution of scalar P is mixing dominated, the
model is adequate. For this reason, the use of equations (42 and 41) as closure model
may yet yield reasonable results.
. A close comparison (of the normalized moments) of the multiscalar mapping closure
model and the multivariate/3-pdf model (Figures 22 and 23) against inert data shows
that the two models are quite close when the initial length scales and diffusivities
19
are similar. The mapping closuremodel is superior whenthoseparametersarevastly
different for the different scalars.
3. The assumed/3-pdfmodel comesreasonablyclosein calculating many of the unnor-
realized momentsof the scalarjoint pdf, evenfor the reacting caseand even when
the initial length scaleand diffusivities of the participating scalarsare quite disparate
(Figures 18- 21). It is the unnormalizedmomentsthat requireclosuremodeling in tur-
bulent calculations. Hence,despitethe disagreementof the normalizedmomentsunder
somecircumstances,the assumed/3-pdfmodelappearsto beadequatefor engineering
calculations.
References
[1] Chen, H., Chen, S. and Kraichnan, R. H. (1989) Probability distribution of a stochasti-
cally advected scalar field. Physical Review Letters, 63 (24), 2657 - 2660.
[2] Eswaran, V., and Pope, S. B. (1988) Direct Numerical Simulations of the turbulent
mixing of a passive scalar. Physics of Fluids, 31 (3), 506 - 520.
[3] Gao, F. (1991) Mapping closure for multispecies Fickian diffusion. Phys. Fluids 3 (10),
2438 - 2444.
[4] Girimaji, S. S. and Pope, S. B. (1990) Material element deformation in isotropic turbu-
lence J. Fluid. Mech. 220, pp 427 - 458.
[5] Girimaji, S. S. (1991a) Assumed 13-pdf model for turbulent mixing: validation and ex-
tension to multiple scalar mixing. Combust. Sci. and Tech. 78, 4 - 6, 177 - 196.
[6] Girimaji, S. S. (1991b) A simple recipe for modding reaction rates in flows with turbulent
combustion. AIAA-91-1792. AIAA 22nd Fluid Dynamics, Plasma Dynamics and Lasers
Conference, June 24-26, 1991, Honolulu, Hawaii.
[7] Girimaji, S. S. (1992a) A mapping closure for turbulent scalar mixing using time-evolving
reference field. Phys. Fluids A (in press).
2O
[8] Girimaji, S. S. (1992b) On the modeling of scalar diffusion in isotropic turbulence. Phys.
Fluids. A, 4 (11), 2529 - 2537.
[9] Girimaji, S. S. (1992c) Towards understanding turbulent scalar mixing. NASA Contract.
Rep. CR 4446.
[10] Girimaji, S. S. (1993) A study ofmultiscalar mixing. Phys. Fluids. A, 5 (7), 1802- 1809.
[11] Madnia, C. K., Frankel, S. H., and Givi, P. (1992) Reactant conversion in homoge-
neous turbulence: mathematical modeling, computational validations and practical ap-
plications. Theoret. Comput. Fluid Dynamics, 4, 79 - 93.
[12] Kerstein, A. R. (1988) A linear eddy model of turbulent scalar transport and mixing.
Combust. Sci. and Tech., 60, 391 - 421.
[13] Pope, S. B. (1985) PDF methods for turbulent reacting flows. Prog. Energy Combust.
Sci., 11, 119- 192.
[14] Pope, S. B. (1991) Mapping closures for turbulent mixing and reaction. Theo. and Comp.
Fluid Dynamics, 2, 255 - 270.
[15] Williams, F. A. (1988) Combustion Theory. Addison - Wesley Publishing Company,
Inc., New York.
21
1.0
Means for Group 1
0.8
0.6 - or_lD._o
g 0 4 "---"-- ,_o u .........
__o oQ-° ._ A_A_A _A
0 o _,A _,,_ A-'_ _'_-J ...
OOAS, A'_A _ 0_o0_0°
O. 2 [- _" = ^0oooo °°°°
8_ 0000000000u
I m e -0 oo0
ooL."oooO°, , , ,0.00 0.01 0.02 0.03 0.04 0.05
Time t
Figure 1: Evolution of the scalar means for Group 1. #a : C1 (solid line), Cla (- -),
Clb (--), Clc(----). #P" Cla (circle), Clb (square), clc(triangle).
co
0.6
0.5
0.4
0.3
Means for Group 2
0.2
0.1
0.00.00
000000000
000 O0
0 oO o
.nO °0
_ _ _ _ _ _ _ 0 O- A_,&AAA&_A&A_°u&
O- AA A_----- 000"-q_ =_ --'-- eq0o°_ Q_- __._ __ ..nO o0-° --.
_o, .. _o_oooO_
m_ _ _00000000 "" _. ... _
@ ^0 o0°
aooOOO/U. 0 , , ,
0.01 0.02 0.03 0.04Time t
,.05
Figure 2: Evolution of the scalar means for Group 2. #a : C2 (solid line), C2a (- -),
C2b (--), C2c(----). #e" C2a (circle), C2b (square), c2c(triangle).
22
Vorionces for Group 10.25
%
0.20
0.15
0.10
0.05
O.OO0.00 0.01 0.02 0.03 0.04 0.05
Time t
Figure 3: Evolution.of the scalar variance for Group 1" Legend same as figure 1
0.25Vorionces for Group 2
0.20
% 0.15_
°°Io.o I .....o.oo___
0.00 0.01 0.02 0,03 0.04 0.05
Time t
Figure 4: Evolution of the scalar variance for Group 2: Legend same as figure 2.
2:3
Correlations0.0
Figure 5:
(square).
-0.2
-0.4
o
-0.6
-0.8 8
- 1.0 - , ...._- u_e'8;djd_:_otx_-_ot_,-_r_,_At_,_r_n_.-_0.00 0.01 0.02 0.03 0.04 0.05
Time t
Evolution of the correlation CAB: C2 (- -), C2b (---), C4 (circle) and C4b
Mean10
color dissipation and chemlcol production rote (Group 1)
%
6
4
2
-20.00
_g88888888888888_lUfiliOoooooeeuoooeooooooo
| I t I
0.01 0.02 0.0,3 0.04 0.05Time t
Figure 6: Mean scalar dissipation and chemical production rate of scalar A for Group 1.
Mean scala," dissipation: C1 (solid line), Cla (--), Clb (--), Clc(----). Chemical
production: Cla (circle), (Jib (square), Clc(triangle).
24
Mean ;color dissipation and chemical production rate (Group 2)10
6
4
%
-2 I i I i0.00 0.01 0.02 0.03 0.04 0.05
Time t
Figure 7: Mean scalar dissipation and chemical production rate of of scalar A for Group
2. Mean scalar dissipation : C2 (solid line), C2a (--), C2b (--), C2c(----). Chemical
production : C2a (circle), C2b (square), C2c(triangle).
1.0
0.8
0.6
• 0.4
0.2
0.0
00000
0
0 0
& AA 0
a 0
/ d'-o i f "_. %\
& 0 _-
/ '_ 0 o ",_
A 0 0
&
_'_,o, __ _ 000000_ _ "
/ noOOOOOOO_OOQoJO0_6__- _ _UOOoo
-0.2 I I I I0.00 0.01 0.02 0.03 0.04 0.05
Figure 8: Mean scalar dissipation and chemical production rate of scalar P for Group 1. Mean
scalar dissipation: C1 (solid line), Cla (--), Clb (--), Clc(----). Chemical production :
Cla (circle), Clb (square), Clc(triangle).
25
Mean scalar dissipation and chemical produclion rote (Group 2)
1.01 0000% 0
0.8 [ o °Oo
I [] O O
I0 6 I- 0 A z, Oo
_o. ' I " " O0
_$ I _ °v- _ .o°..%
/ / _ , %0 ""/ _ I f _ 00%i A_ o
0.21-" / a
II 000000000000_0 0 _'_I /ooooOOo ........ 0o0.0
0.00 0.01 0.02 0.0.3 0.04 0.05Time t
Figure 9: Mean scalar dissipation and chemical production rate of of scalar P for Group2. Mean scalar dissipation : C2 (solid line), C2a (--), C2b (--), C2c(----). Chemical
production: C2a (circle), C2b (square), C2c(triangle).
g
26
10PDF of scalar A (Cases 1)
<
LL
8
4
0"0.0
f
, /k.
J _ I
0.2 0.4
\
__
O.6 O.8 .0
_A
10PDF of scalar A (Cases l b)
<
v
I_1_
8
6
4
2
0
\/
I I _1 I
0.0 0.2 0.4 0.6 0.8 1.0
@A
Figure 10: PDF evolution of the scalar A for C1 (a) and Clb (b): Time t = .006 (- -), 0.012
(----), 0.024 (----) and 0.048 (----).
27
10PDF of scalar A (Case 2)
V
L
00.0 0.2 0.4 0.6 0.8 1.0
_A
10PDF of scalar A (Case 2b)
i,
8
6
4
00.0
ILl
I "-_ -I- _"T_---" _-_ 2._--
0.2 0.4 0.6 0.8 1.0
_A
Figure 11" PDF evolution of the scalar A for C2 (a) and C2b (b): Legend same as figure 10.
28
40
35
30
25
"Z20
U_
15
I
.I
, 1
"t IIII
II
I
I
I
JI
/10
5
00.0
PDF of scalar P (Case lo)
I
II
II
1I
/I
0.2 0.4 0.6 0.8
_p
.0
20
18
16
14
12
Q._-10
b_
8
6
PDF of scalar P (Case l b)
I
I !
T tI II 1
"r _i IiI r I
-_ I I I
J" /I I2,'-'_" _ I I0 "" .I I t I
0.0 0.2 0.6 0.8
II
Ix_ ,-I"
0.4
_P
.0
Figure 12: PDF evolution of the scalar P for Cla (a) and Clb (b): Legend same as figure10.
29
.--&
LL
25
20 -I
PDF of scalar P (Case 2a)
/ 1
i _ 1 I I
0.2 0.4 0.6 0.8
_p
.0
LL
10PDF of scalar P (Case 2b)
8
4
IInIIIIII
L fl 1
tt fl II,i I
2/,/ Ic"
,/I /
0 i l IO.O 0.2 0.4 0.6 0.8
J
.0
Figure 13: PDF evolution of the scalar P for C2a (a) and C2b (b): Legend same as figure
10.
3O
Conditional scalar dissipation of A (Case 2)5
4
_< 3
x2
1"
00.0 0.2 0.4 0.6
_A
0.8 1.0
Conditional scalar dissipation of A (Case 2_)5
4
<3
v
x2
00.0 0.2 0.4 0.6 0.8 1.0
_A
Figure 14: Conditional scalar dissipation of scalar A for C2 (a) and C2b (b): Legend same
as figure 10.
31
Conditional scalar dissipation of P (Case 2a)3.0
2.5
2.0rl
"1o1,3
1.513_
X
i
Ii I/ \1.0
ft/i /L
o._ iII It tI
0.0 L_\ \ , \0.0 0.2
I I I
0.4 O.6 O.8
_#p
.0
Conditional scalar dissipation of P (Case 2b).3.0
\
\,.IX,
0.6 0.8 1.0
Figure 15: Conditional scalar dissipation of scalar P for C2a (a) and C2b (b): Legend same
as figure 10.
32
0.6
0.5
.._ 0.4
---.0.3)<
v0.2t.x_
0.1
0.00.0
I I I I
0.4 0.8
@A
0.6
0.5
#0,4
o.3
_'0.2
0.1
0.00.0 0.4 0.8
_A
<1,3
v
,,.--%
!.a_
0.8
0.7
0.6
0.5
0.4
0.3
0.2%
0.1
0.0'0.0 0.4 O.8
1.0
0.8
_0.6
vX
"_ 0.4LI_
0.2
0.00.0
, i
0.4 0.8
_#A _A
Figure 16: DRS data vs. unmodified mapping closure model. Comparison of I;?((_A)X((_A)/f.A
for Case C2b: Data (solid line), model (circle). (a) t = 0.006, (b) t = 0.012, (c) t = 0.024
and (d) t = 0.048.
33
40
35
50Q.
to
25o_
20
I.L
10
5
00.0
_p
nk)
n
vX
v
8
7
6
5
4
,3
2
1
00.0
£
0.4
(#p
I
0.8
5 5
n
•---, 5n
X
"E2x
LL
00.( 0.4 0.8
r#p
4
n
@
0,0 0.4
_p
I
0.8
Figure 17: DRS data vs, unmodified mapping closure model. Comparison of F(_P)X(_pp)/_ P
for Case C2b: Legend same as figure 16.
34
¢',l¢'4
b
¢q
Comparison
0"25 t
0.20
0.15
_0. I0
0.05
0.000.00
DRS vs. Beto-Pdf model (Cose 1c)
ID13
1313
0.02 0.04
Time t
Corn
0.10
0.05
0.0009
C0
E -0.05oE
-0. I0o%_
-0.15
-0.20
)orison DRS vs. Beta-Pdf model (Case lc)
00000 _-
I I I I
0.02 0.04
Time t
Figure 18: DRS data vs. assumed/3-pdf model. Comparison of various unnormalized mo-
ments for case Clc. (a) (r_: data (solid line), mode] (circ]e); (r._ - data (--), model (square).
(b) (qS, q52) - data (solid line), model (circle); (0_q52)" data (--), mode] (square); and, (qS,&._>
- data (----), model (triangle).
35
Comparison DRS vs. Beta-Pdf model (Case 2c)0.25
0.20
0.15O4
b
tN
_O.lO
0.05
0.000.00
O0_D_ _
0\o
_&OOo\ OO_ _ _ _
OoX__ Uo 0 _
_O0_O000n_ "" ._
_°_Oooo_ooo_o__.,... . -+,_
0.02 0.04
Time t
Comparison DRS vs. Beta-Pdf model (Case 2c)
0"10 k
Figure 19: DRS data vs. assumed fl-pdf model. Comparison of various unnornlalized mo-
tnents for case C2c. Legend same as figure 18.
36
Com)arison DRS vs. Beta-Pdf model (Case 3b)0.25
0.20
0.15b
b'- 0.10
0.05
0.000.00
d_0 _
\
O\
k no \
0 \
k_ I O0 o _ -4
O00 "4 '_ --.
_0000 _ _ .. _.
O.02 O.04
Time t
Com
0.10
0.05
0.00
C
E -0.05oE
-0.10oL_
0-0.15
-0.20
3arison DRS vs. Beta-Pdf model (Case 3b)
,..,OI;]L,_._ _,_O000u_'J .......
BE_'oOOOOOOOUV
-0.25 i i I i0.00 0.02 0.04
Time t
Figure 20: DRS data vs. assumed fl-pdf model. Comparison of various unnormalized mo-
ments for case C3b. Legend same as figure 18.
37
Com )arison DRS vs. Beta-Pdf model (Case 4b)0.25
0.20 -_c_
D\-%
O\
O:x\
_ oo \\
.DDD 0 "_ ". -.0.05 -_ % -- ..
I0.00 I i -a ....0.00 0.04
,..,, 0.15b
t'4
_o.lo
0.02
Time t
Com)arison DRS vs. Beta-Pdf model (Case 4b)0.10
0.05 -
I l l I
0.02 0.04
Time t
0.00orl
-4--'
c(D
E -0.050
E
-0.10690%..
c)-0.15
-0.20
Figure 21: DRS data vs. assumed fl-pdf model. Comparison of various unnormalized mo-
ments for c_se C4b. Legend same as figure 18.
38
0.0
-0.2
-o.4p.4
v_0,6
-0.8
-1.0 i i i0.00 0.06
I I I I I
0.12 0.18
0.0
-0.2
-0.4
_-" -0.6
-0.8
-1.0 I I I0.00 O.OE
I I I I I
0.12 0.18
0.4
0.2
"_ -0.2
-0.4
0.00 0.06
2.0
1.6
•_. 1.2%,,(_ 0.8
0.4
0.0 '0.00
I l I I I I 1
0.12 0.18 0.06 0.12 0.18
Figure 22: Comparison of mapping closure model and fl-pdf model using DRS data for Case
C3. Data (soild line), mapping-closure model (' -) and/3-pdf model (circle).
39
0.0
-0.2
A-0.4
-0.8
0.0( 0.12 0.18
<'_12>
0.0
-0.2
A
-0.4
-o.6v
-0.8
-1.00.00 0.06 0.12 0.18
<1_12>
0.4
0.2
o.o
v -0.2
-0.4
0.00
t
2.0
1.6
A_ 1.2
,,_ 0.8V
0.4
0.0 '0.00
I 1 ] I I
0.06 O.12
<l,_12>
i I I 1 1 I I I I
0.06 0.12 0.18 0.18
<_12>
Figure 2:]: Conlparison of mapping closure model and/3-pdf model using DRS data for Case
C.4. Data (soild line), mapping-closure model (---) and/_-pdf modal (circle).
40
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1. AGENCY USE ONLY(Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED
October 1993 Contractor Report
4, TITLE AND SUBTITLE S. FUNDING NUMBERS
SIMULATIONS OF DIFFUSION-REACTIONEQUATIONS WITH IMPLICATIONS TO TURBULENT C NAS1-19480COMBUSTION MODELING WU 505-90-52-01
i6. AUTHOR(S)
Sharath S. Girimaji
7.
9.
PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
Institute for Computer Applications in Science
and Engineering
Mail Stop 132C, NASA Langley Research Center
Hampton, VA 23681-0001
SPONSORING/MONITORING AGENCY NAME(S) AND AOORESS(ES)
Nation_ Aeronautics and Space Administration
Langley Research Center
Hampton, VA 23681-0001
8. PERFORMING ORGANIZATION
REPORT NUMBER
ICASE Report No. 93- _
10. SPONSORING/MONITORINGAGENCY REPORT NUMBER
NASA CR-191536
ICASE Report No. 93-69
11. SUPPLEMENTARY NOTES
Langley Technical Monitor: Michael F. CardFinal ReportTo be submitted to Theoretical and Computational Fluid Dynamics
12a, DISTRIBUTION/AVAILABILITY STATEMENT
Unclassified-Unlimited
12b. DISTRIBUTION CODE
Subject Category 34
13. ABSTRACT (Maximum 200 words)An enhanced diffusion-reaction reaction system (DRS) is proposed as a statistical model for the evolution of multiplescalars undergoing mixing and reaction in an isotropic turbulence field. The DRS model is close enough to the
scalar equations in a reacting flow that other statistical models of turbulent mixing that decouple the velocity field
from scalar mixing and reaction (e.g. mapping closure model, assumed-pdf models) cannot distinguish the model
equations from the original equations. Numerical simulations of DRS are performed for three scalars evolving from
non-premixed initial conditions. A simple one-step reversible reaction is considered. The data from the simulations
are used (i) to study the effect of chemical conversion on the evolution of scalar statistics, and (ii) to evaluate other
models (mapping-closure model, assumed multivariate fl-pdf model).
\__. _=
14. SUBJECT TERMS
turbulent combustion modeling; scalar mixing; mixing/reaction
17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATIOfi 19. SECURITY CLASSIFICATION
OF REPORT OF THIS PAGE OF ABSTRACT
Unclassified Unclassified[
SN 7540-01-280-5500
_U.S. GOVERNMENT PRINTING OFFICE: 1993 - 528-064/86077
IS. NUMBER OF PAGES
44
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tandard Form298(Rev. 2-89)Prescribedby ANSI Std. Z39 lfl298-102